Amusing answers to tests

For the discussion of math. Duh.

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Gem
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Amusing answers to tests

Postby Gem » Wed Nov 14, 2007 10:03 am UTC

I searched, and I don't think there's a thread like this. Here are some answers to questions that people clearly couldn't do. Maybe you've seen them before, but they're still funny.

Spoiler:
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I think someone has this as their avatar.

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I think my favourites are the elephant, and the dinosaur at the party. Suspiciously raptor-like behaviour there...

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Re: Amusing answers to tests

Postby Hit3k » Wed Nov 14, 2007 12:17 pm UTC

Not really answers to a maths/physics/science test but in my history exam one of the essay questions was "Was hitler considered evil during his dictatorship?. Discuss"[I don't remember the actual question but it was something like that] I wrote one sentence. "Yes, just look at that moustache"
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Re: Amusing answers to tests

Postby Robin S » Wed Nov 14, 2007 12:25 pm UTC

I like how the third guy has correctly spelt "aesthetically" but not "straight".

One of my fellow school students wrote a reasonably amusing answer to a Biology question about swim bladders, but sadly not in a final exam. On a recently-returned piece of coursework, meanwhile, I got a big red note saying "Symbols are not cool" (the coursework consisted of mathematical proofs :|) and, later, "I don't believe this." The latter was apparently due to my methodically proving that the decimal expansion 0.101001000100001000001... was aperiodic, rather than regarding it as obvious.
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Re: Amusing answers to tests

Postby Herman » Wed Nov 14, 2007 4:55 pm UTC

Ha! Funny stuff, Gem.

There's always the hoary, "This problem is trivial and is left as an exercise."

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Re: Amusing answers to tests

Postby SimonM » Wed Nov 14, 2007 5:17 pm UTC

I have a wonderful answer to this, for which this exam paper is too small to hold it.
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Re: Amusing answers to tests

Postby Pesto » Wed Nov 14, 2007 5:23 pm UTC

"A friend of mine" wrote these answers. Probably apocryphal, definately not math, but fun none the less.

Q. What is bravery?
A. This.

As extra credit on a philosophy exam.

Q. Why?
A1. Because.
A2. Why not?

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Re: Amusing answers to tests

Postby 3.14159265... » Wed Nov 14, 2007 8:20 pm UTC

On a real analysis exam, I couldn't prove a theorem, so:

I gave a false proof, hid the mistake in the proof, and wrote under the proof:

"This is a false proof, find the mistake", All I got was a question mark, I am proud lol.
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Re: Amusing answers to tests

Postby Hit3k » Wed Nov 14, 2007 8:51 pm UTC

Pesto wrote:As extra credit on a philosophy exam.

Q. Why?
A1. Because.
A2. Why not?

Speaking of philosophy exams. My friend was telling me about one that contained the question "Why?" and the only person to get it right wrote "Because I can".
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Re: Amusing answers to tests

Postby 3.14159265... » Wed Nov 14, 2007 8:55 pm UTC

Q. What is bravery?
A. This.

This is also the best definition of bravery I have heard.
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Re: Amusing answers to tests

Postby scowdich » Wed Nov 14, 2007 9:04 pm UTC

There's a mythical final exam given by a philosophy professor: he places a chair at the front of the room, passes out blue books, and says "Prove this chair doesn't exist."

Legendarily, the only A was garnered by a student who wrote "What chair?"

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Re: Amusing answers to tests

Postby Mathmagic » Wed Nov 14, 2007 9:13 pm UTC

I always like reading the Is Hell Exothermic or Endothermic? urban legend: http://www.pinetree.net/humor/thermodynamics.html

That's just the first Google hit that came up, but they're pretty much everywhere.
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Re: Amusing answers to tests

Postby Gem » Thu Nov 15, 2007 7:27 am UTC

I've always answered questions properly. I can't help it if I care about my marks! But it would be so much fun to use some of these.

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Re: Amusing answers to tests

Postby arbivark » Thu Nov 15, 2007 8:02 am UTC

My last real job involved being a temp grading papers for high school level state tests.
Fair number of cute drawings - when possible i'd give partial credit.
Ended up having to quit the job when I was instructed to mark as wrong answers that were right, but not the same right answer the teachers were expecting.
Couldn't google the canonical version of the "this is, motherfucker!" joke,but here's:

A novice was trying to fix a broken lisp machine by turning
the power off and on. Knight, seeing what the student was doing spoke
sternly, "You cannot fix a machine by just power-cycling it with no
understanding of what is going wrong." Knight turned the machine off
and on. The machine worked.

citation: http://www.cs.cmu.edu/~mihaib/humor/told.jokes

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Re: Amusing answers to tests

Postby scowdich » Thu Nov 15, 2007 8:40 am UTC

That's from a collection of Zen AI koans, attached to the Jargon File (belonging to Eric S. Raymond). There's some pretty worthy joke material in the jargon file itself, as well.

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Re: Amusing answers to tests

Postby Pesto » Fri Nov 16, 2007 2:55 pm UTC

My college roommate took a really easy math class his last semester in school, just because it would boost his GPA ever so slightly, and he could sit there and mess with the school WiFi. Here's what he just sent me.

It's not much to tell really, and it was a couple of questions that couldn't be answered... They were questions related to distribution tables and basic statistics. We only covered binomial distributions in class, so when I read the first question it struck me as odd. If I remember it correctly, It dealt with coin tosses and the likelihood of getting certain sequences. The problem with this is that a binomial distribution only really works under certain confines. Basically when the number of coin tosses exceeds approximately 100 tosses and the number of outcomes is small (in this case heads or tails) the distribution gets further and further out of wack and begins to converge with the poisson distribution. The first question was answerable based on the information present but the second dealt with getting a specific number of heads given a few thousand coin tosses. This would mean that n is greater than 100 and p=2 which means that we won't get accurate results by using this distribution in favor of using the poisson distribution. I did solve the problem up to where i'd have to look up values from a distribution chart and left that as my answer.

The second question dealt with servers and web pages. They wanted a probability that a web page would be served under a given traffic load. The problem with this is that a server takes a specific amount of time to serve a specific web page and is busy while it is serving that page to a particular individual. You can only measure this using an erlang distribution by measuring the time between incoming web server requests along with the amount of time it takes to serve the bandwidth. This data can be combined to form Erlang Units, which can determine the probability of packet loss/timeouts from the web server. A binomial distribution cannot be used for these purposes because the server is not just "on or off," but rather is tied up whenever it gets a hit which can happen with variable frequency. A binomial distribution just doesn't track enough information. I proved this using formulas I got in the book from a future chapter, and turned it in. I talked to the professor who flat out told me I was wrong until I asked him to look at my response at which he conceded that it was a bad question and that I was indeed correct. I was then asked why the hell I was taking math 104...

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Re: Amusing answers to tests

Postby alphacenturi » Sat Nov 17, 2007 12:25 am UTC

I believe my friend wrote "when you tell me how this will be useful in later life, i'll prove this theorem." it only worked because he is a quasi-genius...

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Re: Amusing answers to tests

Postby JayDee » Sat Nov 17, 2007 2:15 am UTC

It wasn't Maths, but I had a Chemistry exam last week that I might've done really badly. Should've studied. So I amused my self with all the questions I didn't have an answer to.

Like trying to explain reactivity using nothing but poker metaphors. Or writing as much calculation as I could think of, followed by the words 'carry the two' and a random answer.

My favourite was the question on half life. I couldn't remember how to calculate the half life given whatever it was they gave. I could remember how to do the rest, and I'd have to miss out on 10 marks just because I can't remember the first step. So I guessed the answer for part a that would make the rest trivial. That is, part b was how much is gone after 2 seconds, so my answer to a was, 2 seconds. With luck I might get a mark or two out of that question.

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Re: Amusing answers to tests

Postby skeptical scientist » Sat Nov 17, 2007 3:14 am UTC

This isn't really an amusing answer to a test, but rather an amusing story along the lines of the one a couple of posts up.

So, at my undergraduate institution, all students were supposed to take a certain number of courses labeled "writing intensives". A "writing intensive" course was not just a course which involved writing, but also involved the possibility of revision and the professor going to the effort of getting the class certified as a "writing intensive" so they were few and far between, and mostly uninteresting. I took the one writing intensive on the list that interested me, but I had to take a second. Since all of the others looked dreadfully dull, I had already taken several other courses which involved a fair amount of writing which were not "writing intensive" such as Shakespeare, Romantic poetry, and a couple of philosophy courses, I decided that it was a bs requirement, and I would get out of it with as little effort on my part as possible.

As luck would have it, one of the so-called "writing intensive" courses was a course titled "intro to proofs". This class was designed for people interested in majoring in math who had trouble with proof-writing to give them some help before they went to the more advanced math classes where homework assignments consisted entirely of writing proof after proof. I had already taken several such classes, and at the time was enrolled in a pair of graduate math classes. So naturally, I registered for intro to proofs. :roll:

As further luck would have it, one of my roommates was also taking the course, and we cut a deal. She would tell me the assignments and hand in my homework for me. In return, I would give her help if she got stuck on any of the problems. I attended class three days: the first day, the day of the midterm, and the day of the final. I think it was probably the least work any student had ever spent on a writing intensive course. The high point of the class, for me, was the one day, about two-thirds of the way through the semester, when I received a pile of homework back. In my student mailbox. It was delivered to me by inter-office mail, because the professor had despaired of me ever showing up in his class. :P
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Re: Amusing answers to tests

Postby btilly » Sat Nov 17, 2007 3:26 am UTC

I was amused by the student I was grading who solved 16/64 by canceling the 6's. When the prof saw it he said that he wished he had used different numbers so that wouldn't work.

But my favorite test story was at Cal Tech back in the 50's. There was a Russian astrophysicist who gave famously hard tests. So a bunch of junior faculty decided to play a joke on him.

There was a grad student in the class. He took an extra copy of the test. A senior professor came in, and asked him to step out and discuss something. While he was distracted a junior professor came in and took the extra test. They farmed it out to a team of profs, each of which did one question. Then they did the same ruse to get the answers back to the grad student who copied all of the answers in his handwriting and turned it in.

The next day the exams were graded. And the astrophysicist was wandering around telling people (pretend you're hearing this with a thick accent), I have a genius in my class, an absolute genius. He got 50% on one of my tests!
Last edited by btilly on Sat Nov 17, 2007 3:34 am UTC, edited 1 time in total.
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Re: Amusing answers to tests

Postby 3.14159265... » Sat Nov 17, 2007 3:30 am UTC

LOL^^
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Re: Amusing answers to tests

Postby skeptical scientist » Sat Nov 17, 2007 3:52 am UTC

btilly wrote:I was amused by the student I was grading who solved 16/64 by canceling the 6's. When the prof saw it he said that he wished he had used different numbers so that wouldn't work.

lol nice one! Is he sure the student didn't do that intentionally knowing it would work?

And are either of your stories actually true? Because I'm kind of dubious... They're too funny to be true. :)
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Re: Amusing answers to tests

Postby btilly » Sat Nov 17, 2007 4:09 am UTC

skeptical scientist wrote:
btilly wrote:I was amused by the student I was grading who solved 16/64 by canceling the 6's. When the prof saw it he said that he wished he had used different numbers so that wouldn't work.

lol nice one! Is he sure the student didn't do that intentionally knowing it would work?

And are either of your stories actually true? Because I'm kind of dubious... They're too funny to be true. :)


I was the grad student who graded the first one. The course was a calculus course for people who weren't going on at Dartmouth College in the mid 90's. Given other things I saw from that course, I think it really was an honest mistake that ruled out.

I heard the second from a professor who was a grad student there a couple of years after the incident.

So I personally guarantee the first. The second is probably true, but I can't really demonstrate it.
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Re: Amusing answers to tests

Postby mathangelist » Sat Nov 17, 2007 5:21 am UTC

My friend showed me this story the other day, and though it isn't quite down the same line as most of the others, it is pretty awesome (from http://www.mentors.ca/bohr.html).

The following is a question on a physics exam at the University of Copenhagen:

"Describe how to determine the height of a skyscraper with a barometer."
One student replied: "You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building."
This highly original answer so incensed the examiner that he failed the student who immediately appealed on the grounds that his answer was indisputably correct.

The university appointed an independent arbiter to decide the case.

The arbiter ruled that the answer was indeed correct, but did not display any noticeable knowledge of physics. It was decided to call the student in and allow him six minutes in which to provide a verbal answer which showed at least a minimal familiarity with the basic principles of physics.

For five minutes the student sat in silence, forehead creased in thought. The arbiter reminded him that time was running out, to which the student replied that he had several extremely relevant answers, but couldn't make up his mind which to use.

On being advised to hurry up the student replied: "First, you could take the barometer up to the roof of the skyscraper, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from this formula I have worked out for you on my text paper here."

Then the student added, "But, Sir, I wouldn't recommend it. Bad luck on the barometer."

"Another alternative", offered the student, "is this: If the sun is shining you could measure the height of the barometer,then set it on end and measure the length of its shadow. Then you measure the length of the skyscraper's shadow, and thereafter it is a simple matter of proportional geometry to work out the height of the skyscraper. On the paper is the formula for that as well."

"But, Sir, if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in a gravitational formula, which I have determined here this time on a long sheet of paper with a very long and complicated calculation."

"Or, Sir, here's another way, and not a bad one at all. If the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up."

"But if you merely wanted to be very boring and very orthodox about the answer you seem to seek, of course, you could use the barometer to measure the air pressure on the roof, and on the ground, and then convert the difference in millibars into feet to give the height of the building."

"But since we are constantly being exhorted to exercise independence of mind and apply scientific methods, undoubtedly the best way would be to knock on the janitor's door and say to him 'If you would like a nice new barometer, I will give you this one if you tell me the height of this skyscraper'."

The student was Niels Bohr, the only Dane ever to win the Nobel Prize in physics.

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Re: Amusing answers to tests

Postby alphacenturi » Sat Nov 17, 2007 5:27 am UTC

mathangelist: is that true or an urban ledgend?
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Re: Amusing answers to tests

Postby mathangelist » Sat Nov 17, 2007 5:43 am UTC

Must be true, I found it on the internet!

Honestly, who knows....

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Re: Amusing answers to tests

Postby Akira » Sat Nov 17, 2007 6:09 am UTC

I made a reference to a massive nuclear device in a philosophy essay... It went uncommented, though. I was sad.
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Re: Amusing answers to tests

Postby Corporat » Sat Nov 17, 2007 6:46 pm UTC

My biology teacher a couple years ago always made the most creative tests, so in this case, the tests themselves were very amusing. The questions required brainpower, not just memorizing names of the bones, how DNA works, etc. These tests were incredibly hard, and yet he made them fun. For example, on every test he gave away a point on the twelfth question. Our first big test had, "What would you rather be doing other than taking this test? A: Running 10 miles in gym class B: Sticking your head in a beehive C: Chewing broken glass"

Another similar question: "What mutation would you like to have so that you can get 'payback' on [name of teacher]?"
The final exam said "It's the end of the year and I'm tired of making up questions. Write something to amuse me." I think I took 15 minutes of my exam time to draw a picture of an elephant and give it my teacher's name.

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Re: Amusing answers to tests

Postby PaulT » Sat Nov 17, 2007 10:52 pm UTC

The guy officiating when I graduated told the Niels Bohr story in long and excruciating detail. So it must be true - he was wearing a robe!

Anyhow, someone told me about a test they were marking, the question was to find the indefinite integral of 1 dx. One student, having obviously copied someone who'd got it right and not understanding the actual question at all, answered "x + cos n tan t".

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Re: Amusing answers to tests

Postby aguacate » Sun Nov 18, 2007 6:40 am UTC

Once my friend and I turned in our weekly notebook photocopies for our electical engineering class late. As a way of reconciling with the TA, I put a $5 bill on the title page, but I did it pre-photocopy.

Yeah, it was funnier at the time...
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Re: Amusing answers to tests

Postby toughlove » Sun Nov 18, 2007 6:44 am UTC

i heard a story about someone who was doing a physics test, it had something to do with "if the skate board goes down a ramp with an elevation of x degrees, will it reach the platform 2 meters away?"

the person drew an elephant in the skateboards way and wrote

"no it cant, there's an elephant in the middle."
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Re: Amusing answers to tests

Postby hobbesmaster » Sun Nov 18, 2007 7:26 am UTC

toughlove wrote:i heard a story about someone who was doing a physics test, it had something to do with "if the skate board goes down a ramp with an elevation of x degrees, will it reach the platform 2 meters away?"

the person drew an elephant in the skateboards way and wrote

"no it cant, there's an elephant in the middle."


Elephants have also been sighted between the gate and source/drain in MOSFETs, and I'm sure in many other places as well. That picture has been up on collegehumor for quite sometime.

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Re: Amusing answers to tests

Postby bcoblentz » Mon Nov 19, 2007 6:53 pm UTC

On last quarter's EM final I barely knew how to do any of the problems so I wrote a nonsense 60-point bonus question. Not sure if I got any points for it but I passed the class...

I also used the "comments" space on the course evaluation form to write about my professor standing up to murderous pirates after our schooner was boarded. I wish I would have kept it.

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Re: Amusing answers to tests

Postby miraidesuka » Mon Nov 19, 2007 7:20 pm UTC

Not my story, so no credit for me...

When my father was in medical school, he had a professor who was uncannily brilliant. The professor would come in to class with no notes, talk for 80 minutes of the 90 minute class about the most absurdly complex biology ever, and in the last 10 minutes, tie it all together and the entire lecture was crystal clear.

The professor had always advised his students to "answer the question" on all the tests. So the final comes around, and it's one of those evil one question finals that's worth half your grade. The question was something like "Can you...(insert extremely complicated med school biology here)?" Anyway, my dad sits there for about five minutes, writes 'yes' on the paper and leaves. Of course, he gets called into the professor's office to explain how to do it later, but still gets an 'A' for answering the question.
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Re: Amusing answers to tests

Postby 0SpinBoson » Mon Nov 19, 2007 7:25 pm UTC

Since I was told I had to give a quiz the day after the exam to preserve the integrity of the schedule, I had the following question on my Physics II quiz (note that this is electromagnetism without calculus):

Derive, using Maxwell's equations, the speed of light in terms of the fundamental constants e0 and u0.

I got answers ranging from question marks to:

A drawing of an apple tree, with the text "I like apples. I like apples"

"You take the first equation, multiply by a kangaroo and subtract a bowl of soup."

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Re: Amusing answers to tests

Postby Robin S » Mon Nov 19, 2007 9:18 pm UTC

Snopes has debunked the Bohr urban legend, which presumably means that they started it in the first place.
This is a placeholder until I think of something more creative to put here.

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Re: Amusing answers to tests

Postby mister k » Tue Nov 20, 2007 5:46 pm UTC

Pesto wrote:My college roommate took a really easy math class his last semester in school, just because it would boost his GPA ever so slightly, and he could sit there and mess with the school WiFi. Here's what he just sent me.

It's not much to tell really, and it was a couple of questions that couldn't be answered... They were questions related to distribution tables and basic statistics. We only covered binomial distributions in class, so when I read the first question it struck me as odd. If I remember it correctly, It dealt with coin tosses and the likelihood of getting certain sequences. The problem with this is that a binomial distribution only really works under certain confines. Basically when the number of coin tosses exceeds approximately 100 tosses and the number of outcomes is small (in this case heads or tails) the distribution gets further and further out of wack and begins to converge with the poisson distribution. The first question was answerable based on the information present but the second dealt with getting a specific number of heads given a few thousand coin tosses. This would mean that n is greater than 100 and p=2 which means that we won't get accurate results by using this distribution in favor of using the poisson distribution. I did solve the problem up to where i'd have to look up values from a distribution chart and left that as my answer.

The second question dealt with servers and web pages. They wanted a probability that a web page would be served under a given traffic load. The problem with this is that a server takes a specific amount of time to serve a specific web page and is busy while it is serving that page to a particular individual. You can only measure this using an erlang distribution by measuring the time between incoming web server requests along with the amount of time it takes to serve the bandwidth. This data can be combined to form Erlang Units, which can determine the probability of packet loss/timeouts from the web server. A binomial distribution cannot be used for these purposes because the server is not just "on or off," but rather is tied up whenever it gets a hit which can happen with variable frequency. A binomial distribution just doesn't track enough information. I proved this using formulas I got in the book from a future chapter, and turned it in. I talked to the professor who flat out told me I was wrong until I asked him to look at my response at which he conceded that it was a bad question and that I was indeed correct. I was then asked why the hell I was taking math 104...


Dunno about the second question, but the binomial distribution does not get out of whack the larger n gets- it's just the tables don't go that far, and the poisson distribution becomes a better and better estimator that its easier to use the poisson distribution. The correct answer is a binomial one, however. Actually thinking about it considering he's not even talking about samples, merely probabilities, one can accurately calculate the answer......
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gmalivuk
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Re: Amusing answers to tests

Postby gmalivuk » Tue Nov 20, 2007 6:53 pm UTC

mister k wrote:
Pesto wrote:stuff

Dunno about the second question, but the binomial distribution does not get out of whack the larger n gets- it's just the tables don't go that far, and the poisson distribution becomes a better and better estimator that its easier to use the poisson distribution. The correct answer is a binomial one, however. Actually thinking about it considering he's not even talking about samples, merely probabilities, one can accurately calculate the answer......

Yeah, binomial distribution could give you *exact* answers in that case, whereas anything else would be only an approximation. A good approximation, granted, but the binomial distribution formulae never get "inaccurate", they just get harder to implement due to the large number of necessary calculations.
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Re: Amusing answers to tests

Postby McHell » Wed Nov 21, 2007 12:10 am UTC

skeptical scientist wrote:lol nice one! Is he sure the student didn't do that intentionally knowing it would work?

And are either of your stories actually true? Because I'm kind of dubious... They're too funny to be true. :)

I see you are fulfilling your task of being skeptical. But I'm very convinced of the first --- reading it I went "meh, all the frickin time"...

I used to see this at least 20x every year; if there's not the sixes but say 24/10 then they'll rewrite it 24/2.5 and simplify it to 4/5. That's the tragedy of trying to teach math and stats to our 1st year biology students: they are thinking they found the one science study (read: employability) without any maths (read: usage of brain). You explicitely point this out, but over and over... Something with fear of white spaces and hope for partial credits on the one hand and the institute pushing for leniency on the other hand.

Edit:
mathangelist wrote:"But, Sir, if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in a gravitational formula, which I have determined here this time on a long sheet of paper with a very long and complicated calculation."

Gravitational? I don't see how that works. On the other hand, the obvious thing here is that the ideal pendulum's period scales linearly with its length (and EDIT not much else)... Bohr would have got that right.
I heard the same story told by a phys prof, but not attributed to any existing person (it also doesn't get funnier by having happened or not).

Edit2: what on earth was I drinking? Pendulum period scales with length, not mass, how did I decide to write the opposite of what I meant? I blame the hour, too.
Last edited by McHell on Wed Nov 21, 2007 9:51 pm UTC, edited 2 times in total.

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Re: Amusing answers to tests

Postby yelly » Wed Nov 21, 2007 12:21 am UTC

Pesto wrote:Q. What is bravery?
A. This.

This urban legend is prevalent here (Israel) as well, just in the form of a question on a language(?) final exam, and the kid in question passes (I was told this by a teacher who really believed it was true, she even thought the subject in question).
On a different note, my (religious) English teacher once gave us a class assignment (you read right, an unimportant 10 minutes class assignment) which included answering the question "what is man's purpose?", or something along those lines. My friend answered this by explaining that we are carbon based life forms with dicks that have sex. In the same assignment we had to write a prayer-poem, mine ended with the line "you rock my socks". She was not pleased.
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Re: Amusing answers to tests

Postby Phi » Wed Nov 21, 2007 12:22 am UTC

So, these aren't really in lieu of answers to questions I don't know, but I do tend to doodle on my tests. One one of my previous tests in Physics, there was a problem about some child pulling a few blocks tied together by string, and it was asking the tension force between the two blocks. I solved the problem using half of the space, and then used the other half to draw a picture of a monster pulling the blocks, saying, "GRARR! The friction monster is here to screw you over!"

One of my favorite drawings, though, was a small comic. There are two guys talking, and there's a rope hanging in the air. "Dude, check it out, it's my massless rope!" The other guy replies "Oh nice! Did you bring your frictionless pulley?". The next panel involves the two standing there with nothing between them, and guy1 saying "Yeah, but the radius is negligible."

Once, though, I did some very stupid mistake on a test, and ended up with an odd integral. All that I wrote as the answer was "word."


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